In
topology and
mathematics in general, the boundary of a subset of a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
is the set of points in the
closure of not belonging to the
interior of . An element of the boundary of is called a boundary point of . The term boundary operation refers to finding or taking the boundary of a set. Notations used for boundary of a set include
and
. Some authors (for example Willard, in ''General Topology'') use the term frontier instead of boundary in an attempt to avoid confusion with a
different definition used in
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
and the theory of
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
s. Despite widespread acceptance of the meaning of the terms boundary and frontier, they have sometimes been used to refer to other sets. For example, ''Metric Spaces'' by E. T. Copson uses the term boundary to refer to
Hausdorff's border, which is defined as the intersection of a set with its boundary. Hausdorff also introduced the term residue, which is defined as the intersection of a set with the closure of the border of its complement.
A
connected component of the boundary of is called a boundary component of .
Common definitions
There are several equivalent definitions for the of a subset
of a topological space
which will be denoted by
or simply
if
is understood:
- It is the closure of minus the interior of in :
where denotes the closure of in and denotes the
topological interior
In mathematics, specifically in topology,
the interior of a subset of a topological space is the union of all subsets of that are open in .
A point that is in the interior of is an interior point of .
The interior of is the complement of th ...
of in
- It is the intersection of the closure of with the closure of its
complement
A complement is something that completes something else.
Complement may refer specifically to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-class ...
:
- It is the set of points such that every
neighborhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of contains at least one point of and at least one point not of :
A of a set refers to any element of that set's boundary. The boundary
defined above is sometimes called the set's to distinguish it from other similarly named notions such as
the boundary of a
manifold with boundary or the boundary of a
manifold with corners
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
, to name just a few examples.
Properties
The closure of a set
equals the union of the set with its boundary:
where
denotes the
closure of
in
A set is closed if and only if it contains its boundary, and open if and only if it is disjoint from its boundary. The boundary of a set is
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
;
this follows from the formula
which expresses
as the intersection of two closed subsets of
("Trichotomy") Given any subset
each point of
lies in exactly one of the three sets
and
Said differently,
and these three sets are
pairwise disjoint
In mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set.. For example, and are ''disjoint sets,'' while and are not disjoint. A ...
. Consequently, if these set are not empty
[The condition that these sets be non-empty is needed because sets in a partition are by definition required to be non-empty.] then they form a
partition of
A point
is a boundary point of a set if and only if every neighborhood of
contains at least one point in the set and at least one point not in the set.
The boundary of the interior of a set as well as the boundary of the closure of a set are both contained in the boundary of the set.
''Conceptual
Venn diagram showing the relationships among different points of a subset
of
= set of
limit points of
set of boundary points of
area shaded green = set of
interior points of
area shaded yellow = set of
isolated points of
areas shaded black = empty sets. Every point of
is either an interior point or a boundary point. Also, every point of
is either an accumulation point or an isolated point. Likewise, every boundary point of
is either an accumulation point or an isolated point. Isolated points are always boundary points.''
Examples
Characterizations and general examples
The boundary of a set is equal to the boundary of the set's complement:
A set
is a
dense
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
open subset of
if and only if
The interior of the boundary of a closed set is the empty set.
[Let be a closed subset of so that and thus also If is an open subset of such that then (because ) so that (because by definition, is the largest open subset of contained in ). But implies that Thus is simultaneously a subset of and disjoint from which is only possible if ]Q.E.D.
Q.E.D. or QED is an initialism of the Latin phrase , meaning "which was to be demonstrated". Literally it states "what was to be shown". Traditionally, the abbreviation is placed at the end of mathematical proofs and philosophical arguments in pri ...
Consequently, the interior of the boundary of the closure of a set is the empty set.
The interior of the boundary of an open set is also the empty set.
[Let be an open subset of so that Let so that which implies that If then pick so that Because is an open neighborhood of in and the definition of the topological closure implies that which is a contradiction. Alternatively, if is open in then is closed in so that by using the general formula and the fact that the interior of the boundary of a closed set (such as ) is empty, it follows that ]
Consequently, the interior of the boundary of the interior of a set is the empty set.
In particular, if
is a closed or open subset of
then there does not exist any non-empty subset
such that
is also an open subset of
This fact is important for the definition and use of
nowhere dense subsets,
meager subsets, and
Baire space
In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior.
According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are ...
s.
A set is the boundary of some open set if and only if it is closed and
nowhere dense.
The boundary of a set is empty if and only if the set is both closed and open (that is, a
clopen set
In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but their mathematical de ...
).
Concrete examples
Consider the real line
with the usual topology (that is, the topology whose
basis sets are
open interval
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Othe ...
s) and
the subset of rational numbers (whose
topological interior
In mathematics, specifically in topology,
the interior of a subset of a topological space is the union of all subsets of that are open in .
A point that is in the interior of is an interior point of .
The interior of is the complement of th ...
in
is empty). Then
*
*
*
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